In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.
History Polyadic spaces were first studied by S. Mrówka in 1970 as a generalisation of dyadic spaces. The theory was developed further by R. H. Marty, János Gerlits and Murray G. Bell, the latter of whom introduced the concept of the more general centred spaces.
Background A subset K of a topological space X is said to be compact if every open cover of K contains a finite subcover. It is said to be locally compact at a point x ∈ X if x lies in the interior of some compact subset of X. X is a locally compact space if it is locally compact at every point in the space. A proper subset A ⊂ X is said to be dense if the closure Ā = X. A space whose set has a countable, dense subset is called a separable space. For a non-compact, locally compact Hausdorff topological space ( X , τ X ) {\displaystyle (X,\tau _{X})} , we define the Alexandroff one-point compactification as the topological space with the set { ω } ∪ X {\displaystyle \left\{\omega \right\}\cup X} , denoted ω X {\displaystyle \omega X} , where ω ∉ X {\displaystyle \omega \notin X} , with the topology τ ω X {\displaystyle \tau _{\omega X}} defined as follows:
τ X ⊆ τ ω X {\displaystyle \tau _{X}\subseteq \tau _{\omega X}}
X ∖ C ∪ { ω } ∈ τ ω X {\displaystyle X\setminus C\cup \left\{\omega \right\}\in \tau _{\omega X}} , for every compact subset C ⊆ X {\displaystyle C\subseteq X} .
Definition Let X {\displaystyle X} be a discrete topological space, and let ω X {\displaystyle \omega X} be an Alexandroff one-point compactification of X {\displaystyle X} . A Hausdorff space P {\displaystyle P} is polyadic if for some cardinal number λ {\displaystyle \lambda } , there exists a continuous surjective function f : ω X λ → P {\displaystyle f:\omega X^{\lambda }\rightarrow P} , where ω X λ {\displaystyle \omega X^{\lambda }} is the product space obtained by multiplying ω X {\displaystyle \omega X} with itself λ {\displaystyle \lambda } times.
Examples Take the set of natural numbers Z + {\displaystyle \mathbb {Z} ^{+}} with the discrete topology. Its Alexandroff one-point compactification is ω Z + {\displaystyle \omega \mathbb {Z} ^{+}} . Choose λ = 1 {\displaystyle \lambda =1} and define the homeomorphism h : ω Z + → [ 0 , 1 ] {\displaystyle h:\omega \mathbb {Z} ^{+}\rightarrow \left[0,1\right]} with the mapping
h ( x ) = { 1 / x , if x ∈ Z + 0 , if x = ω {\displaystyle h(x)={\begin{cases}1/x,&{\text{if }}x\in \mathbb {Z} +\\0,&{\text{if }}x=\omega \end{cases}}}
It follows from the definition that the image space h [ ω Z ] = { 0 } ∪ { 1 / n : n ∈ N } {\displaystyle h[\omega \mathbb {Z} ]=\left\{0\right\}\cup \left\{1/n\,:\,n\in \mathbb {N} \right\}} is polyadic and compact directly from the definition of compactness, without using Heine-Borel. Every dyadic space (a compact space which is a continuous image of a Cantor set) is a polyadic space. Let X be a separable, compact space. If X is a metrizable space, then it is polyadic (the converse is also true).
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